The Stacks project

Definition 4.27.1. Let $\mathcal{C}$ be a category. A set of arrows $S$ of $\mathcal{C}$ is called a left multiplicative system if it has the following properties:

  1. The identity of every object of $\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$.

  2. Every solid diagram

    \[ \xymatrix{ X \ar[d]_ t \ar[r]_ g & Y \ar@{..>}[d]^ s \\ Z \ar@{..>}[r]^ f & W } \]

    with $t \in S$ can be completed to a commutative dotted square with $s \in S$.

  3. For every pair of morphisms $f, g : X \to Y$ and $t \in S$ with target $X$ such that $f \circ t = g \circ t$ there exists an $s \in S$ with source $Y$ such that $s \circ f = s \circ g$.

A set of arrows $S$ of $\mathcal{C}$ is called a right multiplicative system if it has the following properties:

  1. The identity of every object of $\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$.

  2. Every solid diagram

    \[ \xymatrix{ X \ar@{..>}[d]_ t \ar@{..>}[r]_ g & Y \ar[d]^ s \\ Z \ar[r]^ f & W } \]

    with $s \in S$ can be completed to a commutative dotted square with $t \in S$.

  3. For every pair of morphisms $f, g : X \to Y$ and $s \in S$ with source $Y$ such that $s \circ f = s \circ g$ there exists a $t \in S$ with target $X$ such that $f \circ t = g \circ t$.

A set of arrows $S$ of $\mathcal{C}$ is called a multiplicative system if it is both a left multiplicative system and a right multiplicative system. In other words, this means that MS1, MS2, MS3 hold, where MS1 $=$ LMS1 $+$ RMS1, MS2 $=$ LMS2 $+$ RMS2, and MS3 $=$ LMS3 $+$ RMS3. (That said, of course LMS1 $=$ RMS1 $=$ MS1.)


Comments (2)

Comment #4336 by Manuel Hoff on

In LMS2, the notation doesn't make it entirely clear to me how is quantified. Maybe one wants to mention more explicitly that given and as in the diagram, there exist and as in the diagram (same applies to RMS2).

Comment #4486 by on

Since you parsed the condition correctly, I think others can too. Let's see if other people also feel we should clarify this. (There doesn't see to be a way to dotting the in the diagram.)

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  • 19 comment(s) on Section 4.27: Localization in categories

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